Theorem fusgrvtxfi 27109

Description: A finite simple graph has a finite set of vertices. (Contributed by AV, 16-Dec-2020.)

Hypothesis

Ref	Expression
isfusgr.v	⊢ 𝑉 = (Vtx‘𝐺)

Assertion

Ref	Expression
fusgrvtxfi	⊢ (𝐺 ∈ FinUSGraph → 𝑉 ∈ Fin)

Proof of Theorem fusgrvtxfi

Step	Hyp	Ref	Expression
1		isfusgr.v	. . 3 ⊢ 𝑉 = (Vtx‘𝐺)
2	1	isfusgr 27108	. 2 ⊢ (𝐺 ∈ FinUSGraph ↔ (𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin))
3	2	simprbi 500	1 ⊢ (𝐺 ∈ FinUSGraph → 𝑉 ∈ Fin)

Syntax hints:   → wi 4   = wceq 1538   ∈ wcel 2111  ‘cfv 6324  Fincfn 8492  Vtxcvtx 26789  USGraphcusgr 26942  FinUSGraphcfusgr 27106

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-rab 3115  df-v 3443  df-un 3886  df-in 3888  df-ss 3898  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-iota 6283  df-fv 6332  df-fusgr 27107

This theorem is referenced by:  fusgrfupgrfs  27121  nbfusgrlevtxm1  27167  nbfusgrlevtxm2  27168  nbusgrvtxm1  27169  uvtxnm1nbgr  27194  cusgrm1rusgr  27372  wlksnfi  27693  fusgrhashclwwlkn  27864  clwwlkndivn  27865  fusgreghash2wsp  28123  numclwwlk3lem2  28169  numclwwlk4  28171